Preheat the oven to 350°F.
Heat 1 tablespoon of olive oil and 1 tablespoon of butter in a large saute pan over medium heat. Add the garlic and cook 1 minute until fragrant. Add the scallions and cook 2-3 minutes until softened. Stir in the spinach, parsley, salt, pepper, and nutmeg. Remove from heat and cool. Once mixture is cool, stir in the crumbled feta cheese and egg. Set aside.
To make the bechamel sauce, melt 4 tablespoons of butter in a medium saucepan over medium heat. Add the flour and whisk until smooth. Cook 1-2 minutes, then gradually whisk in the milk. Simmer the sauce until it is thick enough to coat the back of a spoon. Stir in the salt, nutmeg, and pepper. Remove from heat.
To assemble the lasagna, coat the bottom of a 13x9 inch lasagna pan with a thin layer of bechamel sauce. Arrange 4 lasagna noodles along the bottom of the pan, slightly overlapping each other. Pour ⅓ of the bechamel sauce evenly over the noodles, making sure to cover all of the edges. Spread ⅓ of the spinach feta filling evenly over the sauce and top it with 2 tablespoons of Parmesan cheese. Repeat with 2 more layers of noodles, sauce, spinach feta filling, and Parmesan cheese.
To make the phyllo crust, unroll the phyllo dough on a flat surface and keep it covered with a damp towel to prevent it from drying out. Melt the remaining 3 tablespoons of butter in a small bowl. Brush one sheet of phyllo dough with melted butter and sprinkle about 1 teaspoon of breadcrumbs on top. Top with another phyllo sheet, brush with more butter, and sprinkle with more breadcrumbs. Repeat with the remaining phyllo sheets. Once assembled, place the phyllo crust on top of the lasagna and press down to seal the edges. Brush with butter, making sure that the edges are nicely coated to prevent them from curling up during the baking process. Mix the remaining breadcrumbs, Parmesan cheese, and melted butter together and sprinkle the mixture over the top of the lasagna.
Bake the lasagna for 28-30 minutes, being careful not to over-brown the phyllo. Remove from oven and let cool slightly before serving. Cut into wedges and garnish with fresh parsley. Enjoy!